As a consistency check, we compare our result (REF ) with the finding of Ref. [1]} utilizing a crossed-constant field calculation to analyze a high-intensity laser based vacuum birefringence scenario in the parameter regime where \({\rm z}_{\rm R}\ll \tau \) . In this limit and for \(\beta =\pi /4\) , Eq. (REF ) becomes \(\Phi \simeq {\rm z}_{\rm R} \omega (2^{1/4}\alpha /30)(e{\cal E}_0/m_e^2)^2\) , while the corresponding result of Ref. [1]} is \(\Phi _{\text{\cite {Heinzl:2006xc}}}\simeq {\rm z}_{\rm R} \omega (2\alpha /15)(I_0/I_{\rm cr})\) , with peak intensity \(I_0\simeq 2W/(\pi w_0^2\tau )= {\cal E}_0^2\sqrt{\pi /32} \)  and critical intensity \(I_{\rm cr}=(m_e^2/e)^2\) ; cf. also Ref. [3]}. Correspondingly, we find \(\Phi /\Phi _{\text{\cite {Heinzl:2006xc}}}\simeq (2\sqrt{2}/\pi )^{1/2}\approx 0.95\) , which implies an excellent agreement.
